Съдържание

IPhO 1988, Теория и експеримент — Задача 5. EXPERIMENT 2: Electron Tube

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Всички материали от IPhO 1988

19th International Physics Olympiad - 1988 Bad Ischl / Austria

Условие

Introduction

Free electrons in a metal may be thought of as being “electron gas” confined in potential or energy walls. Under normal conditions or even when a voltage is applied near the surface of the metal, these electrons cannot leave the potential walls (see fig. 19.11)

If however the metal or the electron gas is heated, the electrons have enough thermal energy (kinetic energy) to overcome the energy barrier W (W is known as “work function”). If a voltage is applied across the metal and the anode, these thermally activated electrons may reach the anode.

The number of electrons arriving at the anode per unit time depends on the nature of the cathode and the temperature, i.e. all electrons freed from the potential wall will reach the anode no longer increase with applied voltage (see fig. 19.11)

The saturated current corresponding to the number of thermally activated electrons freed from the metal surface per unit time obeys what is generally known as Richardson´s equation i.e.

I_B = C\cdot T^2\cdot e^{-\frac{W}{k\cdot T}}

Energy versus distance diagram for the diode: the cathode well with filled levels ('Cathode'), a sloping potential barrier, and the anode well with filled levels ('Anode'); axes labelled 'Energy' and 'Distance'. — Fig.19.12

where

C is a constant

T temperature of the cathode in Kelvin

k Boltzmann´s constant = $1{,}38\ .\ 10^{-23}$ J/K

Potential well for the electrons in a metal: a horizontal outside level, a rectangular well with a filled level inside, and the work function W marked by arrows between the filled level and the outside level. — Fig 19.11

Graph of current I versus voltage U: three curves rising linearly from the origin and levelling off at three different saturation currents. — Fig 19.13 Graph of current as a function of voltage across anode-cathode

Determine the value of the work function W of tungsten metal in the form of heating filament of the vacuum tube provided.

The following items of equipment are placed at the disposal of the contestants:

Electron tube AZ 41 which is a high-vacuum, full-wave rectifying diode. The cathode is made from a coated tungsten filament the work function of which is to be ascertained. According to the manual prepared by its manufacturer, no more than 4 V should be used when applying heating current to the cathode. Since the tube has two anodes, it is most desirable to have them connected for all measurements. The diagram in fig. 19.14 is a guide to identifying the anodes and the cathode.
multimeter 1 unit, internal resistance for voltage measurement: 10 MΩ
battery 1,5 V (together with a spare)
battery 9 V; four units can be connected in series as shown in fig. 19.15
connectors
resistors; each of which has specifications as follows:
- 1000 Ω ± 2% (brown, black, black, brown, brown, red)
- 100 Ω ± 2% (brown, black, black, black, brown, red)
- 47,5 Ω ± 1% (yellow, violet, green, gold, brown)
resistors; 4 units, each of which has the resistance of about 1 Ω and coded
connecting wires
screw driver
graph paper (1 sheet)
graph of specific resistance of tungsten as a function of temperature; 1 sheet

Base of the tube seen from below: a circle with eight pin holes numbered 1 to 8. — Fig 19.14

Schematic of the tube: an elongated bulb with two anodes a and a′ at the top (labelled 'anodes') and the filament cathode at the bottom (labelled 'cathode'). — Fig 19.14

Four 9 V batteries drawn as rounded rectangles, clipped together in series in two staggered columns. — Fig 19.15

Solve the following problems:

2.1 Determine the resistance of 4 numerically-coded resistors. Under no circumstances must the multimeter be used as an ohmmeter.

2.2 Determine the saturated current for 4 different values of cathode temperatures, using 1,5 V battery to heat the cathode filament. A constant value of voltage between 35 V – 40 V between the anode and the cathode is sufficient to produce a saturated current. Obtain this value of voltage by connecting the four 9 V batteries in series. Describe how the different values of temperature are determined.

2.3 Determine the value of W. Explain the procedures used.

Graph of ρ specific resistance (Ωcm⁻¹) of tungsten versus Temperature (°C): vertical ticks 0.01, 0.02, 0.03, 0.04, horizontal ticks 250, 500, 750, 1000, 1250; the curve rises from about 0.008 at 0 °C almost linearly to about 0.04 at 1250 °C. — Fig 19.16

Решение

2.1

Connect the circuit as shown in fig. 19.17

$R_X$ .... resistance to be determined

R ...... known value of resistance

Measure potential difference across $R_X$ and R. Chose the value of R which gives comparable value of potential difference across $R_X$ .

In this particular case R = 47,5 Ω

\frac{R_X}{R} = \frac{V_X}{V}

Circuit: the unknown resistor R_x and the known resistor R in series with a battery; the voltages V_x across R_x and V across R are indicated by arrows. — Fig. 19.17

where $V_X$ and V are values of potential differences across $R_X$ and R respectively.

$R_X$ can be calculated from the above equation.

(The error in $R_X$ depends on the errors of $V_X$ and $V_R$ ).

2.2

Connect the circuit as shown in fig. 19.18

Circuit: the tube (drawn as a bulb) with an ammeter A and a 9 V battery stack in the anode circuit, and a 1,5 V battery with a variable resistor in the cathode heating circuit. — Fig 19.18

Begin the experiment by measuring the resistance $R_0$ of the tungsten cathode when there is no heating current
Add resistor R = 1000 Ω into the cathode circuit, determine resistance $R_1$ of the tungsten cathode, calculate the resistance of the current-carrying cathode.
Repeat the experiment, using the resistor R = 100 Ω in the cathode circuit, determine resistance $R_2$ of tungsten cathode with heating current in the circuit.
Repeat the experiment, using the resistor R = 47,5 Ω in the cathode circuit, determine resistance $R_3$ of tungsten cathode with heating current in the circuit.
Plot a graph of $\frac{R_1}{R_0}$ , $\frac{R_2}{R_0}$ and $\frac{R_3}{R_0}$ as a function of temperature, put the value of $\frac{R_0}{R_0} = 1$ to coincide with room temperature i.e. 18°C approximately and draw the remaining part of the graph parallel to the graph of specific resistance as a function of temperature provided in the problem. From the graph, read values of the temperature of the cathode $T_1$ , $T_2$ and $T_3$ in Kelvin.

From the equation $I = C\cdot T^2\cdot e^{-\frac{W}{k\cdot T}}$

Graph of specific resistance ρ (Ωcm⁻¹) versus Temperature (°C) with ticks 0.01–0.04 and 250–1250, showing the ρ curve and a parallel curve R/R0 above it; a second vertical scale on the right is labelled R/R0. — Fig 19.19

we get $\ln\frac{I}{T^2} = -\frac{W}{k\cdot T} + \ln C$

Plot a graph of $\ln\frac{I}{T^2}$ against $\frac{1}{T}$ .

The curve is linear. Determine the slope m from this graph. $-m = -\frac{W}{k}$

Work function W can be calculated using known values of m and k (given in the problem).

Error in W depends on the error of T which in turn depends on the error of measured R.

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